Hel${}$lo, will $n$ vectors $(v_i)_i$ iid uniform on $\mathbb{S}^{n-1}$ probably have $|v_i^\dagger v_j|<\varepsilon \forall i\neq j$ for $n $ large? Than${}$ks!
2026-04-04 08:19:27.1775290767
Are $n$ random vectors close to an orthonormal basis?
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No. The Marchenko-Pastur distribution with $\lambda,\sigma^2=1$ describes how skewed such a basis will be.